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Here’s what we have to do to show a coronavirus vaccine works



The moonshot program to come up with a vaccine against covid-19 is advancing faster than anyone could have hoped. At least four experimental vaccines have been shown to protect monkeys, and three of those are already being given to brave human volunteers.

The aim is a vaccine by January, and money is no object. On May 21, the US said it would throw $1.2 billion behind a vaccine from Oxford University and AstraZeneca, part of what President Donald Trump called “Operation Warp Speed.”

It’s all good news, and many scientists believe a vaccine is likely. But the next hurdle is the biggest one: proving that a vaccine candidate actually works.

By early summer, expect to see researchers try to launch several huge efficacy studies involving thousands of volunteers. This is going to be the most costly part of testing a vaccine, and also the hardest to speed up. That’s because researchers will have to wait for people in the study to get accidentally infected with the virus, and then check how often those given the vaccine get sick.

President Trump along with Anthony Fauci, director of the National Institute of Allergies and Infectious Diseases and Secretary of Health and Human Services Alex Azar in March after returning from the NIH’s vaccine research center.

“It’s invariably the most expensive part of vaccine development and the longest of the three phases,” says Stanley Plotkin, former head of vaccines at Sanofi Pasteur. More than anything, he says, the time frame “depends on the incidence of the disease.”

The global effort to create a vaccine on such short notice is unprecedented. New technology has led drug makers to move quickly, and regulators have never pushed paper so fast. But with covid-19 getting beaten back in cities like New York, where new cases have fallen from more than 6,000 a day to less than 600, successful efforts to suppress the disease could perversely make it harder to test a vaccine.

Vaccine makers say it’s a concern. Speaking to Wall Street analysts on a conference call this week, Tal Zaks, chief medical officer at Moderna, whose mRNA vaccine was first to enter human studies in March, said: “The challenge is, how do I ensure I have enough cases? If I go and vaccinate a lot of people, it doesn’t matter how many if there is no circulating virus.”

The irony is that the process will go faster if the covid-19 outbreak keeps flaring up. Vaccine researchers are also expected to pick nurses, doctors, and other at-risk groups to join their studies, so there’s a bigger chance of subjects catching it. They will advise people to stay safe, even while hoping some get sick.

“You still have to tell people to try not to get infected. You don’t say ‘Stop wearing a mask, or ‘Why don’t you meet people in a closed space.’ How is that for a weird dilemma?” says Arthur Caplan, a bioethicist at NYU Langone Health in New York. “The world is trying to get this under control, which I admire, but it does undermine the ability to study a vaccine.”

Trial design

While companies are the ones to decide what a trial looks like, in the US they need approval from the Food and Drug Administration, which has already said it expects to see randomized double-blind trials, the gold standard for proving that a treatment really works.

That means in the big studies starting this summer, some people will get the real vaccine and others will likely get a placebo shot. Then researchers will look to see how many in each group get infected or develop covid-19.

According to Clinton Hermes, an attorney at Bass, Berry & Sims and an outside counsel to the Coalition for Epidemic Preparedness Innovation, a “placebo arm” is necessary to keep both scientists and subjects in the dark. “Knowing who got the vaccine can bias the researcher on a subconscious level in the way they collect data, and bias the behavior of the subject too,” says Hermes. “What you find is that if people are told they were given a vaccine, they are more likely to expose themselves to a virus than if they are uncertain. It’s just human nature to engage in riskier behavior if they think they have been vaccinated.”

Going faster

To run their studies, vaccine companies are likely to look for volunteers among higher-risk populations in big, dense cities, according to Cynthia Dukes, vice president for drug development services at Icon Clinical Research, which is coordinating several covid vaccine and drug trials.

Most people to be vaccinated will likely be first responders, health care workers, or members of the National Guard, she says. “It’s a good population to go into and get an answer. We do want them to have a risk of exposure. If they are sitting at home, it’s not going to work.”

Dukes says she’s seen draft plans for vaccine trials involving 6,000 to 10,000 volunteers, in which researchers estimated that as many as 3.7% of volunteers would be exposed to the virus. That means they’re expecting about 100 to 150 infections in the placebo arm and fewer, or ideally none, among those who get vaccinated. That would be statistical proof the shot works.

During the trials, people will regularly be checked to see if they are infected as well. “In a clinical study, at the first sign of a sniffle you get tested,” says Dukes.

According to Dukes, it is possible to speed up the studies by filling them more quickly. She expects that won’t be a problem; volunteers have been calling in by the thousands. Another way to get them done faster is simply to make them larger. That is what the Department of Health and Human Services signaled it planned to do with the AstraZeneca trial, which it said would involve 30,000 people in the US.

Like a horse race, Operation Warp Speed could end up backing trials of several vaccine contenders. That means the total number of volunteers needed could soar to more than 150,000, Reuters reported. But it’s far from clear there will be enough covid-19 around. Adrian Hill, the Oxford University scientist behind the AstraZeneca shot said he thought there was a 50% chance a trial would lead to “no result at all.”

“It’s a race against the virus disappearing,” Hill told The Telegraph.

The British pharmaceutical company pledged to have 300 million doses ready by October, but some are skeptical that large trials will have delivered answers that soon. “Experts keep saying we’ll have a vaccine in the fall, but we won’t even have any data by then,” says Caplan, who adds that it would be “ethically impossible” to distribute a vaccine before there’s proof it works safely. “People wouldn’t take it. Also, you might be worried about an adverse event that is one in 25,000,” he says. “That’s rare, but if you are vaccinating a billion people, then relatively rare events are going to be more common.”

Anthony Fauci, head of the US National Institutes of Allergy and Infectious Diseases, projected that it will take at least four or five months for a big trial to generate enough evidence that a vaccine works. “If we are successful, we hope to know that in the late fall and early winter,” he told a Senate hearing in May.

Hermes, the vaccine lawyer, agrees that “you won’t see anything deployed in general population without proof of efficacy,” but he thinks governments will be likely to permit emergency use of a vaccine in frontline workers even before all the data is in—perhaps within a few months. “If a vaccine has a good safety profile and might possibly work, you could see people on the front line getting it,” he says.

Immunity correlates

As clues roll in, the whole world is hanging on any hint a vaccine will end the covid crisis. On May 18, Moderna Pharmaceuticals sent the entire US stock market up nearly 4% after it said volunteers who got its vaccine in a safety trial made antibodies at levels similar to convalescent covid-19 patients, a hint that its vaccine might be effective.

In some volunteers, the antibodies were “neutralizing,” meaning they countered the virus when their plasma was tested in a petri dish.

The company was quickly attacked for doing “science by press release,” but Fauci, whose agency is testing the Moderna vaccine, says there is good reason to believe it could work. “It’s definitely not a long shot,” he told Senator Mitt Romney during the Senate hearing. “This is a virus that induces an immune response, and people recover … the very fact that the body is capable of spontaneously clearing the virus tells me that at least from a conceptual standpoint, we can stimulate the body with a vaccine that would induce a similar response.”

That’s a very different situation from HIV—where (with rare exceptions) people never naturally defeat the virus, something that may explain why vaccines for AIDS have flopped time and again. With covid-19, by contrast, about 99 out of 100 people survive and appear to eventually cleared the virus.

The early data from the Moderna trial is an example of what vaccine makers call “correlates of immunity.” A correlate is something they can measure, like antibody levels, and which they can also cause to increase to a certain level with a vaccine. What they need to prove now, in the efficacy trial, is whether or not these factors can predict—or actually cause—real protection against the virus.

“There are lots of kinds of immune responses. So which of these are you creating, and which is responsible for protection, if there is protection? That is the question,” says Plotkin. “Neutralizing antibodies are not always sufficient, but it’s a good guess. For many diseases it’s the key to protection, but not always.”

What could go wrong

What’s extraordinary now is that covid-19 vaccines are advancing to big tests even as the parallel scientific effort to understand our immune response to the coronavirus remains in its early stages. Zaks, the chief medical officer of Moderna, compared the process to “flying a plane” while building it.

And some voices are rising to warn that Operation Warp Speed could easily crash on takeoff. “Normally, it takes up to 10 years to make a vaccine,” Rick Bright, the former head of the US agency responsible for the vaccine push, told lawmakers on May 14. “A lot of optimism is swirling around a 12-to-18-month time frame, if everything goes perfectly. We’ve never seen everything go perfectly.”

So what can go wrong? Dukes, who runs studies on behalf of companies, says that in her experience it’s usually manufacturing. If there’s a problem scaling-up supplies, the process can face a big setback, since everything from animal studies to human studies is supposed to be based on an identical product.

For his part, Plotkin says the usual downfall of a vaccine is that researchers pick the wrong part of the virus to include in it or fail to inject enough of the substance. Other times it’s the biology of the virus that’s the problem—as in the case of HIV, a shape-shifter that attacks the immune system.

It’s also well known that a perfectly effective vaccine can fail to make a difference for a much more mundane reason: people don’t get vaccinated.

That happens more often than you think. Many people don’t bother with flu vaccines, anti-vaccine movements are sowing fear, and there are regions of the world where shots for preventable disease never get delivered.

With covid-19, all those problems remain on the horizon. There’s also the fact that it’s going to be hard to make enough vaccine for everyone. “In the context of a pandemic, we expect demand to far outstrip supply,” Zaks said of his company’s vaccine. During the call with stock analysts, his boss, Stéphane Bancel, the company’s CEO, agreed: “Trust me, it was not part of our business plan to have a billion doses.”

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Keyboard Shortcuts for Calendar, Reminders, and Notes on Mac



If you’re a Mac user looking for a simple and effective day planner, consider this trio of native macOS apps: Calendar, Reminders, and Notes.

Once you set up these apps to your liking, you have a fuss-free system to manage your schedule, tasks, and notes. Plus, if you learn how to control them with keyboard shortcuts, so much the better. And what’s more, you can discover various useful keyboard shortcuts for these macOS productivity apps in the cheat sheet below.

The cheat sheet contains shortcuts for navigation and search, view management, formatting, and more in Calendar, Reminders, and Notes.

FREE DOWNLOAD: This cheat sheet is available as a downloadable PDF from our distribution partner, TradePub. You will have to complete a short form to access it for the first time only. Download Keyboard Shortcuts for Calendar, Reminders, and Notes on Mac.

Keyboard Shortcuts for Calendar, Reminders, and Notes on Mac

Cmd + N Add new event
Option + Cmd + N Add new calendar
Shift + Cmd + N Add new calendar group
Option + Cmd + S Add new calendar subscription
Cmd + F Highlight search box to search for events
Cmd + 1 Switch to Day view
Cmd + 2 Switch to Week view
Cmd + 3 Switch to Month view
Cmd + 4 Switch to Year view
Cmd + Right Arrow Go to next day, week, month, or year
Cmd + Left Arrow Go to previous day, week, month, or year
Cmd + T Switch to today’s date
Shift + Cmd + T Open popup for switching to specific date
Cmd + + (Plus) Increase text size
Cmd + – (Minus) Decrease text size
Cmd + R Refresh all calendars
Cmd + E Edit selected event
Esc (when event is open) Close event editor without saving changes
Return (when event is open) Commit changes to event and close event editor
Cmd + I Show Info popup for selected event(s)
¹Option + Cmd + I Show Inspector popup for selected event
Arrow Keys Select event (if available) in adjacent row/column in relevant direction
Control + Option + Up Arrow Day/Week View: Move selected event 15 minutes earlier
Month View: Move selected event one week earlier
Control + Option + Down Arrow Day/Week View: Move selected event 15 minutes later
Month View: Move selected event one week later
Control + Option + Right Arrow Day/Week/Month View: Move selected event one day later
Control + Option + Left Arrow Day/Week/Month View: Move selected event one day earlier
Shift + Cmd + A Toggle Availability panel
Cmd + N Create new reminder
Shift + Cmd + N Create new list
²Cmd + ] Indent reminder to create subtask
²Cmd + [ Outdent reminder
Cmd + E Show all subtasks
Shift + Cmd + E Hide all subtasks
³Cmd + I Show Info popup for selected reminder
Cmd + F Highlight search box to search for reminders
²Shift + Cmd + F Set/clear flag for selected reminder(s)
Control + Cmd + S Toggle sidebar
Cmd + N Create new note
Shift + Cmd + N Create new folder
Shift + Cmd + A Open dialog for attaching file
Cmd + K Create link
Cmd + F Highlight search box to search current note
Cmd + G Highlight next search result in current note
Shift + Cmd + G Highlight previous search result in current note
Option + Cmd + F Highlight search box to search all notes
Shift + Cmd + T Apply Title format
Shift + Cmd + H Apply Heading format
Shift + Cmd + J Apply Subheading format
Shift + Cmd + B Apply Body format
Shift + Cmd + M Apply Monospaced format
Shift + Cmd + L Apply Checklist format
Shift + Cmd + U Mark selected checklist items as checked/unchecked
Control + Cmd + Up Arrow Move current list/checklist item up in list
Control + Cmd + Down Arrow Move current list/checklist item down in list
Cmd + B Emphasize selected text
Cmd + I Italicize selected text
Cmd + U Underline selected text
Cmd + + (Plus) Increase size of selected text
Cmd + – (Minus) Decrease size of selected text
Cmd + Shift + [ Align selected text flush left
Cmd + Shift + Center selected text
Cmd + Shift + ] Align selected text flush right
Cmd + [ Decrease indent level of selected content or line where cursor is placed
Cmd + ] Increase indent level of selected content or line where cursor is placed
Control + Return Add line break (soft return) to list/checklist item
Option + Tab Insert tab character in list item
Option + Cmd + C Copy style of selection
Option + Cmd + V Paste copied style to selection
Cmd + T Show Fonts window
Shift + Cmd + C Show Colors window
Option + Cmd + T Create table
⁴Return Move cursor to row below
Tab Move focus to next cell on right
Shift + Tab Move focus to next cell on left
Shift + Left/Right Arrow Select cells one by one in relevant direction current row
Shift + Up/Down Arrow Select cells one by one in relevant direction in current column
Option + Return Add new paragraph in current cell
Option + Tab Add tab character in current cell
Option + Cmd + Up Arrow Add new row above current row
Option + Cmd + Down Arrow Add new row below current row
Option + Cmd + Right Arrow Add new column to right of current column
Option + Cmd + Left Arrow Add new column to left of current column
Cmd + 0 Show main Notes window
Cmd + 1 Switch to List view for notes
Cmd + 2 Switch to Gallery view for notes
Cmd + 3 Switch to Attachments Browser
Return (when note is selected in List view or Gallery view) Open or switch focus to selected note to begin typing
Cmd + Return Open or switch focus from current note content to previous notes view i.e. List view or Gallery view
Option+ Cmd + S Toggle Folders sidebar
Shift + Cmd + . (Period) Zoom in on note content
Shift + Cmd + , (Comma) Zoom out of note content
Shift + Cmd + 0 Change note content to default size
⁵Cmd + A (When cursor is in table) Select content of active cell OR
Select table if active cell is empty
Common Shortcuts
Cmd + Z Undo previous action
Shift + Cmd + Z Reverse undo
Cmd + X Cut selected item
Cmd + C Copy selected item
Cmd + V Paste cut/copied item
Delete Delete selected item
Cmd + A Select all items
Cmd + P Open Print dialog
Cmd + , (Comma) Open app preferences
Control + Cmd + F Toggle Full Screen mode
Cmd + M Minimize window
Option + Cmd + M Minimize all windows of current app
⁶Cmd + W Close current window
Option + Cmd + W Close all windows of current app
Cmd + H Hide current app
Option + Cmd + H Hide all apps except current app
Cmd + Q Quit app
¹Shortcut does not work with multiple events, but if you switch between events when Inspector is active, its contents are updated accordingly.

²Shortcut may not be available if iCloud is not enabled.

³If multiple reminders are selected, Info popup for last selected reminder is displayed.

⁴If cursor is in last row, shortcut adds new row to table.

⁵When active cell is populated, press shortcut twice to select table.

⁶In Reminders and Notes, shortcut quits app after closing window.

Bullet Journaling With Mac Productivity Apps

The default productivity apps on macOS are not only easy to use, but also quite flexible. You can use them to bring offline note-taking methods online. For example, you can create a Bullet Journal on your Mac with Calendar, Reminders, or Notes.

Image Credit: Colton Sturgeon on Unsplash

Read the full article: Keyboard Shortcuts for Calendar, Reminders, and Notes on Mac

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If the coronavirus is really airborne, we might be fighting it the wrong way



This was the week airborne transmission became a big deal in the public discussion about covid-19. Over 200 scientists from around the world cosigned a letter to the World Health Organization urging it to take seriously the growing evidence that the coronavirus can be transmitted through the air. WHO stopped short of redefining SARS-CoV-2 (the virus that causes covid-19) as airborne but did acknowledge that more research is “urgently needed to investigate such instances and assess their significance for transmission of COVID-19.”

“I honestly don’t know what people are waiting for,” says microbiologist Chad Roy of Tulane University in the US. “It doesn’t take WHO coming out to make a proclamation that it’s airborne for us to appreciate this is an airborne disease. I don’t know how much clearer it needs to be in terms of scientific evidence.” 

So what does “airborne” really mean in this context? It’s basically an issue of size. We’re pretty sure that SARS-CoV-2 is spread through tiny droplets that contain viral particles capable of leading to an infection. For a virus to be airborne, however, means a few different things, depending on the expert you’re talking to. Typically it means it can spread via inhalation over long distances, perhaps even through different rooms, of small particles known as aerosols.

“That’s why when you ask some of the professionals if the virus is airborne, they’ll say it’s not, because we’re not seeing transmission over those sorts of distances,” says Lisa Brosseau, a retired professor of public health who still consults for businesses and organizations.

There is also some debate on what we mean by “aerosol.” The droplets that carry viral particles through the air can come in all sorts of sizes, but while the larger ones will drop quickly to the ground or other surfaces, the smaller ones (just a few microns across) can linger in the air for a while, giving them a chance to be inhaled. The word is mostly used to describe these smaller particles, although Brosseau would prefer the term “aerosol transmission” to cover the entire gamut of inhalable viral particles being expelled into the air—large and small alike. 

If SARS-CoV-2 is airborne, it’s far from the only disease. Measles is notorious for being able to last in the air for up to two hours. Tuberculosis, though a bacterium, can be airborne for six hours, and Brosseau suggests that coronavirus superspreaders (people who seem to eject a larger amount of the virus than others) disseminate the virus in patterns that recall the infectiousness of tuberculosis.

The evidence that this type of transmission is happening with SARS-CoV-2  arguably already exists. Several big studies point to airborne transmission of the virus as a major route for the spread of covid-19. Other studies have suggested the virus can remain in aerosolized droplets for hours. One new study led by Roy and his team at Tulane shows that infectious aerosolized particles of SARS-CoV-2 could actually linger in the air for up to 16 hours, and maintain infectivity much longer than MERS and SARS-CoV-1 (the other big coronaviruses to emerge this century). 

We still don’t know what gives SARS-CoV-2 this airborne edge. “But it may be one reason this is a pandemic, and not simply a small outbreak like any other coronavirus,” says Roy. 

How to stay safe

Whether the virus is airborne isn’t simply a scientific question. If it is, it could mean that in places where the virus has not been properly contained (e.g., the US), the economy needs to be reopened more slowly, under tighter regulations that reinforce current health practices as well as introducing improved ones. Our current tactics for stopping the spread won’t be enough.

Roy would like to see aggressive mandates on strict mask use for anyone leaving home. “This virus sheds like crazy,” he says. “Masking can do an incredible amount in breaking transmission. I think anything that can promote the use of masking, to stop the production of aerosols in the environment, would be helpful.” 

Brosseau, however, says that though masks can limit the spread of larger particles, they are less helpful for smaller ones, especially if they fit only loosely. “I wish we would stop relying on the idea that face coverings are going to solve everything and help flatten the curve,” she says. “It’s magical thinking—it’s not going to happen.” For masks to really make a difference, they would need to be worn all the time, even around family.

Brosseau does believe the evidence is trending toward the conclusion that airborne transmission is “the primary and possibly most important mode of transmission for SARS-CoV-2.” She says, “I think the amount of time and effort devoted to sanitizing every single surface over and over and over again has been a huge waste of time. We don’t need to worry so much about cleaning every single surface we touch.” Instead, the focus should be on other factors, like where we spend our time.

Crowded spaces

One of the biggest questions we still have about covid-19 is how much of a viral load is needed to cause infection. The answer changes if we think it is aerosols that we need to worry about. Smaller particles won’t carry as large a viral load as bigger ones, but because they can linger in the air for much longer, it may not matter—they’ll build up in larger concentrations and get distributed more widely the longer an infected person is around to expel aerosolized virus. 

The more people you have coming in and out of an indoor space, the more likely it is that someone who is infected will show up. The longer those infected individuals spend in that space, the higher the concentration of virus in the air over time. This is particularly bad news for spaces where people congregate for hours on end, like restaurants, bars, offices, classrooms, and churches. 

Airborne transmission doesn’t necessarily mean these places must stay closed (although that would be ideal). But wiping down surfaces with disinfectant, and having everyone wear masks, won’t be enough. To safely reopen, these spots will not just need to reduce the number of people allowed inside at any given moment; they will also need to reduce the amount of time those people spend there. Increasing social distancing beyond six feet would also help keep people safer. 

Ventilation needs to be a higher priority too. This is going to be a big problem for older buildings that usually have worse ventilation systems, and areas with a lot of those might need to remain closed for much longer. The impact of asymptomatic spread (transmission by people who don’t feel ill) and superspreaders only compounds the problem even further. But research conducted by the US Department of Homeland Security has shown that in the presence of UV light, aerosolized particles of the size the Tulane researchers studied would disappear in less than a minute. A number of businesses have begun deploying UV-armed robots to disinfect hospital rooms, shopping malls, stores, public transit stations, and more.

For many places, considerable delays in economic reopening might ultimately be the price of getting the virus under control. Otherwise the kind of thing that happened when a single open bar in Michigan led to an outbreak of more than 170 new cases could become commonplace. 

For Brosseau, the best strategy is simply to behave as we did in the early days of lockdown—stay home, and avoid coming into contact with anyone you don’t live with. And if you have to leave home, she says, “all I can say is spend as little time as possible in an enclosed space, in an area that’s well ventilated, with as few people as possible.”

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Mmm… Obfuscated Shell Donuts



In case you grow tired of clear-written, understandable code, obfuscation contests provide a nice change of scenery, and trying to make sense of their entries can be a fun-time activity and an interesting alternative to the usual brainteasers. If we ever happen to see a Simpsons episode on the subject, [Andy Sloane] has the obvious candidate for a [Hackerman Homer] entry: a rotating ASCII art donut, formatted as donut-shaped C code.

The code itself actually dates back to 2006, but has recently resurfaced on Reddit after [Lex Fridman] posted a video about it on YouTube, so we figured we take that chance to give some further attention to this nifty piece of art. [Andy]’s blog article goes in all the details of the rotation math, and how he simply uses ASCII characters with different pixel amounts to emulate the illumination. For those who prefer C over mathematical notation, we added a reformatted version after the break.

Sure, the code’s donut shape is mainly owed to the added filler comments, but let’s face it, the donut shape is just a neat little addition, and the code wouldn’t be any less impressive squeezed all in one line — or multiple lines of appropriate lengths. However, for the actual 2006 IOCCC, [Andy] took it a serious step further with his entry, and you should definitely give that one a try. For some more obfuscated shell animations, check out the fluid dynamics simulator from a few years back, and for a more recent entry, have a look at the printf Tic Tac Toe we covered last month.

int k;
double sin();
double cos();

main() {
  float A=0;
  float B=0;
  float i;
  float j;
  float z[1760];
  char  b[1760];


  for (;;) {
    memset(b, 32, 1760);
    memset(z, 0, 7040);

    for (j = 0; 6.28 > j; j += 0.07) {
      for (i = 0; 6.28 > i; i += 0.02) {
        float c = sin(i);
        float d = cos(j);
        float e = sin(A);
        float f = sin(j);
        float g = cos(A);
        float h = d + 2;
        float D = 1 / (c * h * e + f * g + 5);
        float l = cos(i);
        float m = cos(B);
        float n = sin(B);
        float t = c * h * g - f * e;

        int x = 40 + 30 * D * (l * h * m - t * n);
        int y = 12 + 15 * D * (l * h * n + t * m);
        int o = x + 80 * y;
        int N = 8 * ((f * e - c * d * g) * m - c * d * e - f * g - l * d * n);

        if (22 > y && y > 0 && x > 0 && 80 > x && D > z[o]) {
          z[o] = D;
          b[o] = ".,-~:;=!*#$@"[N > 0 ? N : 0];

    for (k = 0; 1761 > k; k++) {
      putchar(k % 80 ? b[k] : 10);

    A += 0.04;
    B += 0.02;

If you want to slow down (or speed up) the animation, decrease (or increase) the values added to A and B at the very end of the loop. Keep them in the same proportion to retain the rotation animation, or just play around with them and see what happens.

Remember to link against the Math library with -lm when compiling.

[via /r/programming]

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